On Coefficient Estimates For New Subclasses of

Special Issue-(2nd ICTSA-2022) International Journal of Innovative Science and Research Technology
ISSN No:-24562165
On Coefficient Estimates for New Subclasses of

-Bi-Spirallike Functions
Read. S. A. Qahtan
College Of Engineering
Alrowad University
Taiz, Yemen
Abstract:- In this paper, we introduce and investigate two for functions in these two new subclasses
new subclasses of the function class of - - - for functions.
spirallike functions defined in the open unit disc.
Furthermore, We find estimates on the coefficients Keywords:- Univalent Functions, -Univalent Functions, -
-Spirallike, Subordination, Coefficients Bounds.
I. INTRODUCTION
Let denote the class of functions of the form
Which are analytic in the open disc . Let denote the subclass of function in which are
univalent in and indeed normalized by It is well known that every function has an inverse
defined by
and
A function is said to bi-univalent function in if and are together univalent functions in . Let denote
the class of bi-univalent functions defined in . The inverse function is given by
Spacek [22] introduced the concept of spirallikeness which is a natural generalization of starlikeness. Spirallike functions
can be characterized by the following analytic condition:
A function in is -spirallik if and only if,
Where In [11], Jackson introduced and studied the concept of the -derivative operator as follows :
Equivalently (4), may be written as
Where , note that as
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ISSN No:-24562165
 Definition 1.1 Let - - denote the class of - -bi-spirallike functions of order . The
function , given by (1), is said it is in - - if it satisfies:
and
(7)
 2 Main Results
 Theorem 2.1 Let

Be in - - Then
Where
Proof. Let
is analytic in and satisfies and It can be checked that the function defined
by:
Is a member of the class .

ISSN No:-24562165
Let
By comparing coefficient in (9), we have
Where
Similarly we take
Where is Analytic in and Satisfies
The function defined by
Is a Member of the class . Let
By comparing coefficient in (13), we have
Where
From (10) and (14) we have

ISSN No:-24562165
We shall obtain a refined estimate on for use in the estimates of and For this purpose we first add (11) with
(15), then use the relations (17) and get the following:
Putting from (10) we have after simplification:
By applying the familiar inequalities and we get:
and
We next find a bound on . For this we substract (15) from (11) and get
The relation from (17), reduces the above expression to
Using and (18), we get

ISSN No:-24562165
Where
Therefore, the inequalities and give the following:
Which Simplifies
Now we find an estimate on . At first we shall derive relation connecting and . To this end, Now we
collect (12) and (16) we get
Where
Now we are putting in (21) we get
Where
Substituting from(20) in (21) we get after simplification:
Since , we have
SINCE , WE HAVE

ISSN No:-24562165
Where
Or
Observing that we have and therefore
We Replace
By the right hand side of (22) ,
put
and
This gives

ISSN No:-24562165
Where
Next, replacing by the expression in the right hand side of (23) and by (18) we finally get
Where
This Gives
Where

ISSN No:-24562165
Where
By applying the inequalities we get
Where

ISSN No:-24562165
As in the above Theorem we get the following:
 Corollary 2.1 [21] Let be in - Then
 Theorem 2.2 Let , given by (1) in the class . Then
and
Proof. Let , then by Definition 1.1 we have
and

ISSN No:-24562165
Where
and
As in the proof of Theorem 2.1, by suitably comparing coefficient in (31) and (32) we have
Where and
Where
In order to express interms of and we first add (34) and (37) and get
Again putting from (33) we have
Or equivalently
The familiar inequalities yield
Which Implies that

ISSN No:-24562165
Following the Lines of Proof of Theorem 2.1, with Appropriate Changes, we Get that
The inequalities yield
We shall next find an estimate on , By substracting (38) from (35) we get
A substitution of the value of from the relation (33) gives
Therefore, using the inequalities , , the estimate for from (41)and the estimate for
from (42), we get
Or equivalently,
As in the above Theorem we get the following:
 Corollary 2.2 [21] Let , given by (1) in the class . then

ISSN No:-24562165
II. CONCLUSIONS [14]. W.C.Ma, D.Minda, A unified treatment of some

special classes of functions, in: Proceedings of the
In this paper, we introduced and investigated two new Conference on Complex Analysis, Tianjin (1992),
subclasses of the function class of - - -spirallike 157-169.
functions defined in the open unit disc. Furthermore, We find [15]. A.K.Mishra and M.M.Soren, "Coefficient bounds for
estimates on the coefficients for functions in bi-starlike analytic functions." Bulletin of the Belgian
these two new subclasses for functions. Future work making Mathematical Society-Simon Stevin,21(1)(2014), 157-
use of the values of a2 a3 and a4 we can caluculate Hankel 167.
determinant coefficient for the bi- spirallike function classes. [16]. G.Murugusundaramoorthy, Subordination results for
spirallike functions associated with Hurwitz-Lerch
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2(2012), 97-103.
[1]. I.Aleksandar, The Riemann Zeta-Function: Theory [17]. G.Murugusundaramoorthy and T.janani, Bi- starlike
and Applications, John-Wiley and Sons, Inc., New function of complex order associated with double zeta
York (1985). functions, Afrika Matematika online June(2014).
[2]. E.W.Barnes, The theory of the double gamma [18]. C.Pommerenke, Univalent Functions. Vandenhoeck
function, Philos. Trans. Roy. Soc.,196(1901), 265-387. and Ruprecht,G¨ottingen (1975).
[3]. M.G.Bin-Saad, Hypergeometric seires assotiated with [19]. W.Rogosinski, On the coefficients of subordinate
the Hurwitz-Lerch zeta function, Acta Math. Univ. functions, Proc. London Math. Soc.,48(1943), 48-82.
Comenianae Vol.,2(2009), 269- 286. [20]. S.Siregar, S.Raman, Certain subclasses of analytic and
[4]. D.A.Brannan, T.S.Taha, On some classes of bi- bi-univalent functions involving double zeta functions,
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Math.,31(2)(1986), 70-77. 16-18.
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associated with the Hurwitz-Lerch zeta function, Appl. Bi-spirallike Analytic Functions. Kyungpook Math.
Math. Comput.,170(2005), 399-409. J.,58(4)(2018).
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satisfying subordinate conditions, J. Classical Pest. Math. Fys.,62(1933), 12-19.
Anal.,2(2013), 49-60. [23]. H.M.Srivastava, A.K.Mishra, P.Gochhayat, Certain
[7]. A.Erdelyi, W.Magnus, F.Oberhettinger, F.G.Tricomi, subclasses of analytic and bi-univalent functions,
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Hill, New York, Toronto and London (1953).
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Special Issue-(2nd ICTSA-2022) International Journal of Innovative Science and Research Technology

Let denote the class of functions of the form

A function in is -spirallik if and only if,

Equivalently (4), may be written as

Where , note that as

IJISRT22DEC1495 www.ijisrt.com 105

 Theorem 2.1 Let

Is a member of the class .

IJISRT22DEC1495 www.ijisrt.com 106

By comparing coefficient in (9), we have

Where is Analytic in and Satisfies

The function defined by

Is a Member of the class . Let

By comparing coefficient in (13), we have

From (10) and (14) we have

IJISRT22DEC1495 www.ijisrt.com 107

Putting from (10) we have after simplification:

By applying the familiar inequalities and we get:

The relation from (17), reduces the above expression to

Using and (18), we get

IJISRT22DEC1495 www.ijisrt.com 108

Therefore, the inequalities and give the following:

Now we are putting in (21) we get

Substituting from(20) in (21) we get after simplification:

IJISRT22DEC1495 www.ijisrt.com 109

Observing that we have and therefore

By the right hand side of (22) ,

IJISRT22DEC1495 www.ijisrt.com 110

IJISRT22DEC1495 www.ijisrt.com 111

By applying the inequalities we get

IJISRT22DEC1495 www.ijisrt.com 112

As in the above Theorem we get the following:

 Corollary 2.1 [21] Let be in - Then

 Theorem 2.2 Let , given by (1) in the class . Then

Proof. Let , then by Definition 1.1 we have

IJISRT22DEC1495 www.ijisrt.com 113

Again putting from (33) we have

The familiar inequalities yield

Which Implies that

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The inequalities yield

We shall next find an estimate on , By substracting (38) from (35) we get

A substitution of the value of from the relation (33) gives

As in the above Theorem we get the following:

 Corollary 2.2 [21] Let , given by (1) in the class . then

IJISRT22DEC1495 www.ijisrt.com 115

II. CONCLUSIONS [14]. W.C.Ma, D.Minda, A unified treatment of some

IJISRT22DEC1495 www.ijisrt.com 116

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